A title that caught my attention on arXiv: testing MCMC code by Roger Grosse and David Duvenaud. The paper is in fact a tutorial adapted from blog posts written by Grosse and Duvenaud, on the blog of the Harvard Intelligent Probabilistic Systems group. The purpose is to write code in such a modular way that (some) conditional probability computations can be tested. Using my favourite Gibbs sampler for the mixture model, they advocate computing the ratios

to make sure they are exactly identical. (Where x denotes the part of the parameter being simulated and z anything else.) The paper also mentions an older paper by John Geweke—of which I was curiously unaware!—leading to another test: consider iterating the following two steps:

update the parameter θ given the current data x by an MCMC step that preserves the posterior p(θ|x);

update the data x given the current parameter value θ from the sampling distribution p(x|θ).

Since both steps preserve the joint distribution p(x,θ), values simulated from those steps should exhibit the same properties as a forward production of (x,θ), i.e., simulating from p(θ) and then from p(x|θ). So with enough simulations, comparison tests can be run. (Andrew has a very similar proposal at about the same time.) There are potential limitations to the first approach, obviously, from being unable to write the full conditionals [an ABC version anyone?!] to making a programming mistake that keep both ratios equal [as it would occur if a Metropolis-within-Gibbs was run by using the ratio of the joints in the acceptance probability]. Further, as noted by the authors it only addresses the mathematical correctness of the code, rather than the issue of whether the MCMC algorithm mixes well enough to provide a pseudo-iid-sample from p(θ|x). (Lack of mixing that could be spotted by Geweke’s test.) But it is so immediately available that it can indeed be added to every and all simulations involving a conditional step. While Geweke’s test requires re-running the MCMC algorithm altogether. Although clear divergence between an iid sampling from p(x,θ) and the Gibbs version above could appear fast enough for a stopping rule to be used. In fine, a worthwhile addition to the collection of checkings and tests built across the years for MCMC algorithms! (Of which the trick proposed by my friend Tobias Rydén to run first the MCMC code with n=0 observations in order to recover the prior p(θ) remains my favourite!)

After several clones of our SAME algorithm appeared in the literature, it is rather fun to see another paper acknowledging the connection. SAME but different was arXived today by Zhao, Jiang and Canny. The point of this short paper is to show that the parallel implementation of SAME leads to efficient performances compared with existing standards. Since the duplicated latent variables are independent [given θ] they can be simulated in parallel. They further assume independence between the components of those latent variables. And finite support. As in document analysis. So they can sample the replicated latent variables all at once. Parallelism is thus used solely for the components of the latent variable(s). SAME is normally associated with an annealing schedule but the authors could not detect an improvement over a fixed and large number of replications. They reported gains comparable to state-of-the-art variational Bayes on two large datasets. Quite fun to see SAME getting a new life thanks to computer scientists!

In a comment on our Accelerating Metropolis-Hastings algorithms: Delayed acceptance with prefetching paper, Philip commented that he had experimented with an alternative splitting technique retaining the right stationary measure: the idea behind his alternative acceleration is again (a) to divide the target into bits and (b) run the acceptance step by parts, towards a major reduction in computing time. The difference with our approach is to represent the overall acceptance probability

and, even more surprisingly than in our case, this representation remains associated with the right (posterior) target!!! Provided the ordering of the terms is random with a symmetric distribution on the permutation. This property can be directly checked via the detailed balance condition.

In a toy example, I compared the acceptance rates (acrat) for our delayed solution (letabin.R), for this alternative (letamin.R), and for a non-delayed reference (letabaz.R), when considering more and more fractured decompositions of a Bernoulli likelihood.

A very interesting outcome since the acceptance rate does not change with the number of terms in the decomposition for the alternative delayed acceptance method… Even though it logically takes longer than our solution. However, the drawback is that detailed balance implies picking the order at random, hence loosing on the gain in computing the cheap terms first. If reversibility could be bypassed, then this alternative would definitely get very appealing!

This new arXival by Chris Oates, Mark Girolami, and Nicolas Chopin (warning: they all are colleagues & friends of mine!, at least until they read those comments…) is a variation on control variates, but with a surprising twist namely that the inclusion of a control variate functional may produce a sub-root-n (i.e., faster than √n) convergence rate in the resulting estimator. Surprising as I did not know one could get to sub-root-n rates..! Now I had forgotten that Anne Philippe and I used the score in an earlier paper of ours, as a control variate for Riemann sum approximations, with faster convergence rates, but this is indeed a new twist, in particular because it produces an unbiased estimator.

The control variate writes

where π is the target density and φ is a free function to be optimised. (Under the constraint that πφ is integrable. Then the expectation of ψφ is indeed zero.) The “explanation” for the sub-root-n behaviour is that ψφ is chosen as an L2 regression. When looking at the sub-root-n convergence proof, the explanation is more of a Rao-Blackwellisation type, assuming a first level convergent (or presistent) approximation to the integrand [of the above form ψφ can be found. The optimal φ is the solution of a differential equation that needs estimating and the paper concentrates on approximating strategies. This connects with Antonietta Mira’s zero variance control variates, but in a non-parametric manner, adopting a Gaussian process as the prior on the unknown φ. And this is where the huge innovation in the paper resides, I think, i.e. in assuming a Gaussian process prior on the control functional and in managing to preserve unbiasedness. As in many of its implementations, modelling by Gaussian processes offers nice features, like ψφ being itself a Gaussian process. Except that it cannot be shown to lead to presistency on a theoretical basis. Even though it appears to hold in the examples of the paper. Apart from this theoretical difficulty, the potential hardship with the method seems to be in the implementation, as there are several parameters and functionals to be calibrated, hence calling for cross-validation which may often be time-consuming. The gains are humongous, so the method should be adopted whenever the added cost in implementing it is reasonable, cost which evaluation is not clearly provided by the paper. In the toy Gaussian example where everything can be computed, I am surprised at the relatively poor performance of a Riemann sum approximation to the integral, wondering at the level of quadrature involved therein. The paper also interestingly connects with O’Hagan’s (1991) Bayes-Hermite [polynomials] quadrature and quasi-Monte Carlo [obviously!].

Approximate Bayesian computation techniques are 2000’s successors of MCMC methods as handling new models where MCMC algorithms are at a loss, in the same way the latter were able in the 1990’s to cover models that regular Monte Carlo approaches could not reach. While they first sounded like “quick-and-dirty” solutions, only to be considered until more elaborate solutions could (not) be found, they have been progressively incorporated within the statistican’s toolbox as a novel form of non-parametric inference handling partly defined models. A statistically relevant feature of those ACB methods is that they require replacing the data with smaller dimension summaries or statistics, because of the complexity of the former. In almost every case when calling ABC is the unique solution, those summaries are not sufficient and the method thus implies a loss of statistical information, at least at a formal level since relying on the raw data is out of question. This forced reduction of statistical information raises many relevant questions, from the choice of summary statistics to the consistency of the ensuing inference.

In this paper of the special MCMSki 4 issue of Statistics and Computing, Stoehr et al. attack the recurrent problem of selecting summary statistics for ABC in a hidden Markov random field, since there is no fixed dimension sufficient statistics in that case. The paper provides a very broad overview of the issues and difficulties related with ABC model choice, which has been the focus of some advanced research only for a few years. Most interestingly, the authors define a novel, local, and somewhat Bayesian misclassification rate, an error that is conditional on the observed value and derived from the ABC reference table. It is the posterior predictive error rate

integrating in both the model index m and the corresponding random variable Y (and the hidden intermediary parameter) given the observation. Or rather given the transform of the observation by the summary statistic S. The authors even go further to define the error rate of a classification rule based on a first (collection of) statistic, conditional on a second (collection of) statistic (see Definition 1). A notion rather delicate to validate on a fully Bayesian basis. And they advocate the substitution of the unreliable (estimates of the) posterior probabilities by this local error rate, estimated by traditional non-parametric kernel methods. Methods that are calibrated by cross-validation. Given a reference summary statistic, this perspective leads (at least in theory) to select the optimal summary statistic as the one leading to the minimal local error rate. Besides its application to hidden Markov random fields, which is of interest per se, this paper thus opens a new vista on calibrating ABC methods and evaluating their true performances conditional on the actual data. (The advocated abandonment of the posterior probabilities could almost justify the denomination of a paradigm shift. This is also the approach advocated in our random forest paper.)

This recently arXived paper by Juho Kokkala and Simo Särkkä mixes a whole lot of interesting topics, from particle MCMC and Rao-Blackwellisation to particle filters, Kalman filters, and even bear population estimation. The starting setup is the state-space hidden process models where particle filters are of use. And where Andrieu, Doucet and Hollenstein (2010) introduced their particle MCMC algorithms. Rao-Blackwellisation steps have been proposed in this setup in the original paper, as well as in the ensuing discussion, like recycling rejected parameters and associated particles. The beginning of the paper is a review of the literature in this area, in particular of the Rao-Blackwellized Monte Carlo Data Association algorithm developed by Särkkä et al. (2007), of which I was not aware previously. (I alas have not followed closely enough the filtering literature in the past years.) Targets evolve independently according to Gaussian dynamics.

In the description of the model (Section 3), I feel there are prerequisites on the model I did not have (and did not check in Särkkä et al., 2007), like the meaning of targets and measurements: it seems the model assumes each measurement corresponds to a given target. More details or an example would have helped. The extension against the existing appears to be the (major) step of including unknown parameters. Due to my lack of expertise in the domain, I have no notion of the existence of similar proposals in the literature, but handling unknown parameters is definitely of direct relevance for the statistical analysis of such problems!

The simulation experiment based on an Ornstein-Uhlenbeck model is somewhat anticlimactic in that the posterior on the mean reversion rate is essentially the prior, conveniently centred at the true value, while the others remain quite wide. It may be that the experiment was too ambitious in selecting 30 simultaneous targets with only a total of 150 observations. Without highly informative priors, my beotian reaction is to doubt the feasibility of the inference. In the case of the Finnish bear study, the huge discrepancy between priors and posteriors, as well as the significant difference between the forestry expert estimations and the model predictions should be discussed, if not addressed, possibly via a simulation using the posteriors as priors. Or maybe using a hierarchical Bayes model to gather a time-wise coherence in the number of bear families. (I wonder if this technique would apply to the type of data gathered by Mohan Delampady on the West Ghats tigers…)

Overall, I am slightly intrigued by the practice of running MCMC chains in parallel and merging the outcomes with no further processing. This assumes a lot in terms of convergence and mixing on all the chains. However, convergence is never directly addressed in the paper.

Here is the third set of slides for my third year statistics course. Nothing out of the ordinary, but the opportunity to link statistics and simulation for students not yet exposed to Monte Carlo methods. (No ABC yet, but who knows?, I may use ABC as an entry to Bayesian statistics, following Don Rubin’s example! Surprising typo on the Project Euclid page for this 1984 paper, by the way…) On Monday, I had the pleasant surprise to see Shravan Vasishth in the audience, as he is visiting Université Denis Diderot (Paris 7) this month.